Impulsive fractional functional differential equations

$Page 1: Impulsive fractional functional differential equations$
Computers and Mathematics with Applications 64 (2012) 3414–3424

Contents lists available at SciVerse ScienceDirect

Computers and Mathematics with Applications

journal homepage: www.elsevier.com/locate/camwa

Impulsive fractional functional differential equations✩

Tian Liang Guo, Wei Jiang ∗

School of Mathematical Sciences, Anhui University, Hefei, Anhui 230039, PR China

a r t i c l e i n f o

Keywords:Fractional functional differential equationImpulseExistenceUniquenessData dependence

a b s t r a c t

In this paper, a class of impulsive fractional functional differential equations is investigated.The first purpose is introducing a natural formula of solutions for impulsive fractionalfunctional differential equations. The second purpose is establishing related new existence,uniqueness and data dependence results. Examples are given to illustrate the results.

© 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Impulsive differential equations have played an important role in modelling phenomena, especially in describingdynamics of populations subject to abrupt changes as well as other phenomena such as harvesting, diseases, and so forth,some authors have used impulsive differential systems to describe the model since the last century. For the basic theoryon impulsive differential equations, the reader can refer to the monographs of Burton and Simeonov [1], Lakshmikanthamet al. [2] and Benchohra et al. [3].

Recently, fractional differential equations have been proved to be valuable tools in the modelling of many phenomena invarious fields of engineering, physics and economics. It draws a great application in nonlinear oscillations of earthquakes,many physical phenomena such as seepage flow in porous media and in fluid dynamic traffic models. Actually, fractionaldifferential equations are considered as an alternative model to integer differential equations. For more details on fractionalcalculus theory, one can see the monographs of Diethelm [4], Kilbas et al. [5], Lakshmikantham et al. [6], Miller and Ross [7],Michalski [8], Podlubny [9] and Tarasov [10]. Fractional differential equations (inclusion) and control problems involving theRiemann–Liouville fractional derivative or the Caputo fractional derivative have been paidmore andmore attention (see forexample [11–22]).

In addition, some modelling is done via impulsive functional differential equations when these processes involvehereditary phenomena such as biological and socialmacrosystems. For fractional functional differential equations, the initialvalue problem (IVP for short), for a class of nonlinear fractional functional differential equations are discussed in [23,24],existence and uniqueness results for fractional neutral differential equations with infinite delay are considered in [14,22],existence and uniqueness results for p-type fractional neutral differential equations are investigated in [21]. For impulsivefractional differential equations involving the Caputo fractional derivative, the existence and uniqueness of solutions for theIVP for nonlinear impulsive fractional differential equations are discussed in [25], the existence of solutions for a class of theIVP for impulsive fractional differential equations are investigated in [26] by using themethod of upper and lower solutions.In particular, a pioneering work on IVP for nonlinear impulsive fractional evolution equations and the Ulam stability fornonlinear fractional differential equations has been reported byWang et al. [27–30]. To the best of our knowledge, impulsivefractional functional differential equations involving the Caputo fractional derivative have not been studied very perfectly.

It is worth remarking that Feckan et al. [31] give a counterexample to show that the formula of solutions in previouspapers are incorrect and reconsider a class of impulsive fractional differential equations and introduce a correct formula of

✩ This work is supported by National Natural Science Foundation of China (11071001).∗ Corresponding author.

E-mail addresses: [email protected] (T.L. Guo), [email protected] (W. Jiang).

0898-1221/$ – see front matter© 2011 Elsevier Ltd. All rights reserved.doi:10.1016/j.camwa.2011.12.054

http://dx.doi.org/10.1016/j.camwa.2011.12.054

http://www.elsevier.com/locate/camwa

http://www.elsevier.com/locate/camwa

mailto:[email protected]

mailto:[email protected]

http://dx.doi.org/10.1016/j.camwa.2011.12.054

T.L. Guo, W. Jiang / Computers and Mathematics with Applications 64 (2012) 3414–3424 3415

solutions for a impulsive Cauchy problemwith Caputo fractional derivative. Further, some sufficient conditions for existenceof the solutions are established by applying fixed point methods.

In this paper, we consider the existence, uniqueness and data dependence of the solutions for IVP of the followingfractional functional differential equation with impulse.

cDq0,tx(t) :=

c Dqt x(t) = f (t, xt), t ∈ J ′ := J \ {t1, . . . , tm} , J := [0, T ],

1x(tk) := x(t+k )− x(t−k ) = Ik(x(t−k )), k = 1, 2, . . . ,m,x|[−τ ,0] = ϕ,

(1)

where cDqt is the Caputo fractional derivative, 0 = t0 < t1 < · · · < tm < tm+1 = T , f : J×PC1

→ Rn is Lebesguemeasurablewith respect to t on J and f (t, φ) is continuous with respect to φ on PC1, Ik : Rn

→ Rn are continuous for k = 1, . . . ,m,where PC1

= PC1([−τ , 0], Rn) denotes the space of piecewise left continuous derivative functions ϕ : [−τ , 0] → Rn withthe sup-norm ∥ϕ∥1 = sup−τ≤s≤0 ∥ϕ(s)∥, here τ is a positive constant, ∥ · ∥ is a suitable complete norm in Rn. xt ∈ PC1 isdefined by xt(s) = x(t + s) for −τ ≤ s ≤ 0. x(t+k ) = limϵ→0+ x(tk + ϵ) and x(t−k ) = limϵ→0− x(tk + ϵ) represent the rightand left limits of x(t) at t = tk. Set PC1(ρ) = {ϕ ∈ PC1

: ∥ϕ∥1 ≤ ρ} for all ρ > 0.The rest of this paper is organized as follows. In Section 2, we give some notations and recall some concepts and

preparation results. Moreover, we introduce a suitable definition of solutions for the system (1) and give a very importantequivalent result. In Section 3, the existence and uniqueness of solution for the system (1) is showed by virtue of fractionalcalculus, Schauder fixed point theorem and Banach contraction principle. In Section 4, we discuss data dependence ofsolutions for the system (1). At last, some examples are given to demonstrate the application of our main results.

2. Preliminaries

In this section, we introduce notations, definitions, and preliminary facts. Throughout this paper, denote byPC1([−τ , T ], Rn) the Banach space of all piecewise continuous functions from [−τ , T ] into Rn which have left continuousderivative on J with the norm

∥x∥∞ = supt∈[−τ ,T ]

∥x(t)∥,

where ∥ · ∥ is a suitable complete norm in Rn. Set PC1([−τ , T ])(ρ) = {x ∈ PC1([−τ , T ], Rn) : ∥x∥∞ ≤ ρ}, ∀ρ > 0.Let us recall the following known definitions. For more details see [5].

Definition 2.1. The fractional integral of order γ with the lower limit zero for a function f : [0,∞) → R is defined as

Iγt f (t) =1

0(γ )

t

0

f (s)(t − s)1−γ

ds, t > 0, γ > 0,

provided the right side is point-wise defined on [0,∞), where 0(·) is the gamma function.

Definition 2.2. The Riemann–Liouville derivative of order γ with the lower limit zero for a function f : [0,∞) → R can bewritten as

LDγt f (t) =1

0(n − γ )

dn

dtn

t

0

f (s)(t − s)γ+1−n

ds, t > 0, n − 1 < γ < n.

Definition 2.3. The Caputo derivative of order γ for a function f : [0,∞) → R can be written as

cDγt f (t) =LDγt

f (t)−

n−1k=0

tk

k!f (k)(0)

, t > 0, n − 1 < γ < n.

Remark 2.4. (i) If f (t) ∈ Cn[0,∞), then

cDγt f (t) =1

0(n − γ )

t

0

f (n)(s)(t − s)γ+1−n

ds = In−γt f (n)(t), t > 0, n − 1 < γ < n.

(ii) The Caputo derivative of a constant is equal to zero.

Lemma 2.5 ([24]). Let 0 < q < 1 and let f : J × PC1→ Rn be continuous, the IVP

cDqt x(t) = f (t, xt), t ∈ [0, T ],

x|[−τ ,0] = ϕ,(2)

3416 T.L. Guo, W. Jiang / Computers and Mathematics with Applications 64 (2012) 3414–3424

is equivalent to the following Volterra fractional integral with memory

x(t) =

ϕ(t), t ∈ [−τ , 0],

ϕ(0)+1

0(q)

t

0(t − s)q−1f (s, xs)ds, t ∈ [0, T ].

(3)

That is, every solution of (3) is also a solution of (2) and vice versa.

Lemma 2.6 ([31]). Let q ∈ (0, 1) and h : J → R be continuous. A function x ∈ C(J, R) is a solution of the fractional integralequation

x(t) = x0 −1

0(q)

a

0(a − s)q−1h(s)ds +

10(q)

t

0(t − s)q−1h(s)ds,

if and only if x is a solution of the following fractional Cauchy problemcDqt x(t) = h(t), t ∈ J,

x(a) = x0, a ∈ (0, T ).(4)

Lemma 2.7. Let 0 < q < 1 and let f : J × PC1→ Rn be Lebesgue measurable with respect to t on J and f (t, φ) is continuous

with respect to φ on PC1. A function x ∈ PC1([−τ , T ], Rn) is a solution of the IVP (1) if and only if x ∈ PC1([−τ , T ], Rn) is asolution of the following fractional integral equation

x(t) =

ϕ(t), for t ∈ [−τ , 0],

ϕ(0)+1

0(q)

t

0(t − s)q−1f (s, xs)ds, for t ∈ [0, t1],

ϕ(0)+ I1(x(t−1 ))+1

0(q)

t

0(t − s)q−1f (s, xs)ds, for t ∈ (t1, t2],

...

ϕ(0)+

mk=1

Ik(x(t−k ))+1

0(q)

t

0(t − s)q−1f (s, xs)ds, for t ∈ (tm, T ].

(5)

Proof. Assume x satisfies impulsive problem (1). If t ∈ [0, t1] then from Lemma 2.5,

x(t) = ϕ(0)+1

0(q)

t

0(t − s)q−1f (s, xs)ds. (6)

and

x(t−1 ) = ϕ(0)+1

0(q)

t1

0(t1 − s)q−1f (s, xs)ds.

If t ∈ (t1, t2] thencDq

t x(t) = f (t, xt), t ∈ (t1, t2] with1x(t1) = I1(x(t−1 )).

By Lemma 2.6, one obtain

x(t) = x(t+1 )−1

0(q)

t1

0(t1 − s)q−1f (s, xs)ds +

10(q)

t

0(t − s)q−1f (s, xs)ds

= x(t−1 )+ I1(x(t−1 ))−1

0(q)

t1

0(t1 − s)q−1f (s, xs)ds +

10(q)

t


= ϕ(0)+ I1(x(t−1 ))+1

0(q)

t

0(t − s)q−1f (s, xs)ds.

If t ∈ (t2, t3] then again using Lemma 2.6, we get

x(t) = x(t+2 )−1

0(q)

t2

0(t2 − s)q−1f (s, xs)ds +

10(q)

t


= x(t−2 )+ I2(x(t−2 ))−1

0(q)

t2

0(t2 − s)q−1f (s, xs)ds +

10(q)

t


= ϕ(0)+ I1(x(t−1 ))+ I2(x(t−2 ))+1

0(q)

t

0(t − s)q−1f (s, xs)ds.


If t ∈ (tm, T ] then again from Lemma 2.6, we get

x(t) = ϕ(0)+

mk=1

Ik(x(t−k ))+1

0(q)

t

0(t − s)q−1f (s, xs)ds.

Conversely, assume that x satisfies (5). If t ∈ [0, t1] then using the fact that cDqt is the left inverse of Iqt we get (6). If

t ∈ (tk, tk+1], k = 1, 2, . . . ,m then using the fact of the Caputo derivative of a constant is equal to zero, we obtaincDq

t x(t) = f (t, xt), t ∈ (tk, tk+1] and1x(tk) = Ik(x(t−k )). This complete the proof. �

Theorem 2.8 (Hölder’s Inequality). Assume that σ , p ≥ 1, and 1σ

+1p = 1. If l ∈ Lσ (J),m ∈ Lp(J), then for 1 ≤ p ≤ ∞, lm ∈

L1(J) and ∥lm∥L1(J) ≤ ∥l∥Lσ (J)∥m∥Lp(J).

For measurable functionsm : J → R, define the norm

∥m∥Lp(J) =

J|m(t)|pdt

1p

, 1 ≤ p < ∞,

infµ(J)=0

{ supt∈J−J

|m(t)|}, p = ∞,

where µ(J) is the Lebesgue measure on J . Let Lp(J, R) be the Banach space of all Lebesgue measurable functions m : J → Rwith ∥m∥Lp(J) < ∞.

Theorem 2.9 (Lemma 2.4, [1]). Let X be a Banach space and W ⊂ PC(J, X). If the following conditions are satisfied:

(i) W is uniformly bounded subset of PC(J, X);(ii) W is equicontinuous in (tk, tk+1), k = 0, 1, 2, . . . ,m, where t0 = 0, tm+1 = T ;(iii) W (t) = {u(t) | u ∈ W , t ∈ J \ {t1, . . . , tm}},W (t+k ) = {u(t+k ) | u ∈ W } and W (t−k ) = {u(t−k ) | u ∈ W } are relatively

compact subsets of X.Then W is a relatively compact subset of PC(J, X).

3. Existence and uniqueness results

In this section, we consider the existence and uniqueness for impulsive fractional functional differential equation (1).For x ∈ PC1([−τ , T ], Rn), we now define a mapping as follow

(Ax)(t) =

ϕ(t), for t ∈ [−τ , 0],

ϕ(0)+1

0(q)

t

0(t − s)q−1f (s, xs)ds, for t ∈ [0, t1],

ϕ(0)+ I1(x(t−1 ))+1

0(q)

t

0(t − s)q−1f (s, xs)ds, for t ∈ (t1, t2],

...

ϕ(0)+

mk=0

Ik(x(t−k ))+1

0(q)

t

0(t − s)q−1f (s, xs)ds, for t ∈ (tm, T ].

(7)

Before stating and proving the result, we introduce the following hypotheses.

[H1]: There exist a constant β ∈ (0, q) and a real-valued function m(t) ∈ L1β (J) such that ∥f (t, φ)∥ ≤ m(t), for almost

every t ∈ J and all φ ∈ PC1.[H2]: There exist a constant γ ∈ (0, q) and a real-valued function µ(t) ∈ L

1γ (J) such that ∥f (t, φ) − f (t, ψ)∥ ≤

µ(t)∥φ − ψ∥1, for almost every t ∈ J and all φ,ψ ∈ PC1.[H3]: Ik ∈ C(Rn, Rn) and maps bounded set to bounded set. There exists a constant λ∗ > 0 such that ∥Ik(x(t−k )) −

Ik(y(t−k ))∥ ≤ λ∗∥x − y∥∞ for each x, y ∈ PC1([−τ , T ])(ρ) and k = 1, 2, . . . ,m.

Theorem 3.1. Assume [H1] hold. Then the IVP (1) has at least one solution on J, provided that

∥ϕ(0)∥ +1

0(q)

1−βq−β

1−βT q−βM + mM∗

ρ≤ 1, (8)

where M = ( T0 (m(s))

1β ds)β , M∗

= max{∥Ik(x(t−k ))∥ : ∥x∥∞ ≤ ρ}, k = 1, . . . ,m.


Proof. By the definition of PC1([−τ , T ])(ρ), it is easy to know that PC1([−τ , T ])(ρ) is a closed, bounded and convex subsetof PC1([−τ , T ], Rn).

We shall use the Schauder fixed point theorem to prove that A defined by (7) has a fixed point. The proof will be given inseveral steps.

Step 1. A maps PC1([−τ , T ])(ρ) into PC1([−τ , T ])(ρ).

To prove this we need use the Hölder inequality (Theorem 2.8). Direct calculation gives that (t − s)q−1∈ L

11−β [0, t], for

t ∈ J . In light of the Hölder inequality and the condition [H1], we obtain that (t − s)q−1f (s, xs) is Lebesgue integrable withrespect to s ∈ [0, t], and t

0∥(t − s)q−1f (s, xs)∥ds ≤

t

0((t − s)q−1)

11−β ds

1−β t

0(m(s))

1β ds

β. (9)

According to the inequality (8) and Hölder inequality (9), for t ∈ [0, t1], we have

∥(Ax)(t)∥ ≤ ∥ϕ(0)∥ +1

0(q)

t

0(t − s)q−1m(s)ds

≤ ∥ϕ(0)∥ +1

0(q)

t

0((t − s)q−1)

11−β ds

1−β t

0(m(s))

1β ds

β≤ ∥ϕ(0)∥ +

10(q)

1 − β

q − β

1−β

tq−β T

0(m(s))

1β ds

β≤ ∥ϕ(0)∥ +

10(q)

1 − β

q − β

1−β

tq−β1 M

≤ ρ. (10)

Similarly, for t ∈ (tk, tk+1], k = 1, . . . ,m, we have

∥(Ax)(t)∥ ≤ ∥ϕ(0)∥ +

ki=1

Ii(x(t−i ))

+1

0(q)

t

0(t − s)q−1m(s)ds

≤ ∥ϕ(0)∥ + mM∗+

10(q)

1 − β

q − β

1−β

T q−βM

≤ ρ. (11)

Combining (10) and (11), and note that ∥(Ax)(t)∥ = ∥ϕ(t)∥ ≤ ∥ϕ∥1 ≤ ρ, for t ∈ [−τ , 0], it yields

∥Ax∥∞ ≤ ρ.

Therefore, A : PC1([−τ , T ])(ρ) → PC1([−τ , T ])(ρ).

Step 2. A is continuous.Let {xl} be a sequence such that xl → x on PC1([−τ , T ])(ρ). This, together with the continuity of f (t, φ) with respect

to φ and Ik(ψ) with respect to ψ , implies that f (t, xlt) → f (t, xt), t ∈ J and Ik(xl) → Ik(x) on PC1([−τ , T ])(ρ) asl → ∞, k = 1, 2, . . . ,m.

For each t ∈ [0, t1],

∥(Axl)(t)− (Ax)(t)∥ =1

0(q)

t

0(t − s)q−1

[f (s, xls)− f (s, xs)]ds

≤tq1

0(q + 1)sups∈J

∥f (s, xls)− f (s, xs)∥. (12)

Similarly, for each t ∈ (tk, tk+1], k = 1, . . . ,m

∥(Axl)(t)− (Ax)(t)∥ =1

0(q)

t

0(ti − s)q−1

[f (s, xls)− f (s, xs)]ds+

ki=1

∥Ii(xl(t−i ))− Ii(x(t−i ))∥

≤T q

0(q + 1)sups∈J

∥f (s, xls)− f (s, xs)∥ +

ki=1

∥Ii(xl)− Ii(x)∥. (13)


Since, as l → ∞, f (t, xlt) is convergent to f (t, xt), t ∈ J and Ik(xl) is convergent to Ik(x), k = 1, . . . ,m, combining (12) and(13) and note that ∥(Axl)(t)− (Ax)(t)∥ = ∥ϕ(t)− ϕ(t)∥ ≡ 0, for t ∈ [−τ , 0], it yields

∥Axl − Ax∥∞ → 0 as l → ∞,

this follows that A is continuous.

Step 3. A maps bounded sets into equicontinuous sets of PC1([−τ , T ], Rn).Let PC1([−τ , T ])(ρ) be a bounded set as in Step 1 and 2, and x ∈ PC1([−τ , T ])(ρ).It is obviously that Ax is equicontinuous on the time interval [−τ , 0].For arbitrary s1, s2 ∈ [0, t1], s1 < s2, based on the Hölder inequality (Theorem 2.8), we obtain

∥(Ax)(s2)− (Ax)(s1)∥

=1

0(q)

s1

0[(s2 − s)q−1

− (s1 − s)q−1]f (s, xs)ds +

s2

s1(s2 − s)q−1f (s, xs)ds

≤

10(q)

s1

0∥[(s2 − s)q−1

− (s1 − s)q−1]f (s, xs)∥ds +

10(q)

s2

s1∥(s2 − s)q−1f (s, xs)∥ds

≤1

0(q)

s1

0[(s1 − s)q−1

− (s2 − s)q−1]m(s)ds +

10(q)

s2

s1(s2 − s)q−1m(s)ds

≤1

0(q)

s1

0[(s1 − s)q−1

− (s2 − s)q−1]

11−β ds

1−β s1

0(m(s))

1β ds

β+

10(q)

s2

s1[(s2 − s)q−1

]1

1−β ds1−β s2

s1(m(s))

1β ds

β≤

10(q)

s1

0

(s1 − s)

q−11−β − (s2 − s)

q−11−βds1−β T

0(m(s))

1β ds

β+

10(q)

s2

s1

(s2 − s)

q−11−βds1−β T

0(m(s))

1β ds

β≤

M0(q)

1 − β

q − β

1−β sq−β1−β1 + (s2 − s1)

q−β1−β − s

q−β1−β2

1−β+

M0(q)

1 − β

q − β

1−β (s2 − s1)

q−β1−β1−β

≤2M0(q)

1 − β

q − β

1−β

(s2 − s1)q−β .

As s2 → s1, the right-hand side of the above inequality tends to zero. Then Ax is equicontinuous on interval [0, t1].In general, for the time interval (tk, tk+1], we similarly obtain the following inequality

∥(Ax)(s2)− (Ax)(s1)∥ ≤2M0(q)

1 − β

q − β

1−β

(s2 − s1)q−β → 0, as s2 → s1.

This yields that Ax is equicontinuous on (tk, tk+1] for k = 1, . . . ,m.On the other hand, because that A(PC1([−τ , T ]))(ρ) ⊂ PC1([−τ , T ])(ρ) is uniformly bounded according to the

conclusion derived in Step 1, then applying to PC-typeAscoli–Arzela (Theorem2.9 in the case ofX = Rn),A(PC1([−τ , T ]))(ρ)is a relatively compact subset of PC1([−τ , T ], Rn). Thus A : PC1([−τ , T ])(ρ) → PC1([−τ , T ])(ρ) is completely continuous.

As a consequence of Steps 1–3 together with the Schauder fixed point theorem, we deduce that A has a fixed point inPC1([−τ , T ])(ρ)which is a solution of the IVP (1) on J . The proof is complete. �

[H1′] There exist a positive constant L such that ∥f (t, φ)∥ ≤ L, for almost every t ∈ J and all φ ∈ PC1(ρ).

Corollary 3.2. Assume [H1′] hold, then the IVP (1) at least has a solution on J, provided that

∥ϕ(0)∥ +1

0(q+1)TqL + mM∗

ρ≤ 1.

Proof. We just need to prove that A maps PC1([−τ , T ])(ρ) into PC1([−τ , T ])(ρ). The other part of the proof is similar toTheorem 3.1.


In light of the Hölder inequality and the condition [H1′], for t ∈ [0, t1], we have

∥(Ax)(t)∥ ≤ ∥ϕ(0)∥ +L

0(q)

t

0(t − s)q−1ds

≤ ∥ϕ(0)∥ +1

0(q + 1)tq1L

≤ ρ. (14)


∥(Ax)(t)∥ ≤ ∥ϕ(0)∥ +

ki=1

Ii(x(t−i ))

+L

0(q)

t

0(t − s)q−1ds

≤ ∥ϕ(0)∥ + mM∗+

10(q + 1)

T qL

≤ ρ. (15)

Combining (14) and (15) and note that ∥(Ax)(t)∥ = ∥ϕ(t)∥ ≤ ∥ϕ∥1 ≤ ρ, for t ∈ [−τ , 0], it yields

∥Ax∥∞ ≤ ρ.

Therefore, A : PC1([−τ , T ])(ρ) → PC1([−τ , T ])(ρ). This complete the proof. �

Theorem 3.3. Assume [H1]–[H3] hold, then the IVP (1) has a unique solution on J, provided that the inequality (8) and thefollowing inequality

c :=1

0(q)

1 − γ

q − γ

1−γ

T q−γµ∗+ mλ∗ < 1, (16)

hold, where µ∗= (

T0 (µ(s))

1γ ds)γ .

Proof. Let A be the function defined by (7). Then A : PC1([−τ , T ])(ρ) → PC1([−τ , T ])(ρ) is well defined according toTheorem 3.1.

Next, we shall use the Banach contraction principle to prove the A has a fixed point.For arbitrary x1, x2 ∈ PC1([−τ , T ])(ρ) and t ∈ [−τ , 0],

∥(Ax1)(t)− (Ax2)(t)∥ = ∥ϕ(t)− ϕ(t)∥ ≡ 0. (17)

For t ∈ [0, t1], from the conditions [H2]–[H3], the Hölder inequality (9) and the inequality (16), we have

∥(Ax1)(t)− (Ax2)(t)∥ ≤1

0(q)

t

0(t − s)q−1

[f (s, x1s )− f (s, x2s )]ds

≤1

0(q)

t

0(t − s)q−1µ(s)ds sup

s∈[0,t]∥x1s − x2s ∥1

≤1

0(q)

t

0(t − s)q−1µ(s)ds∥x1 − x2∥∞

≤1

0(q)

t

0((t − s)q−1)

11−γ ds

1−γ t

0(µ(s))

1γ ds

γ∥x1 − x2∥∞

≤1

0(q)

1 − γ

q − γ

1−γ

tq−γ1 µ∗∥x1 − x2∥∞

≤ c∥x1 − x2∥∞. (18)


∥(Ax1)(t)− (Ax2)(t)∥ ≤1

0(q)

t

0(t − s)q−1

[f (s, x1s )− f (s, x2s )]ds+

ki=1

∥Ii(x1(t−i ))− Ii(x2(t−i ))∥

≤1

0(q)

t


s∈[0,t]∥x1s − x2s ∥1 + kλ∗

∥x1 − x2∥∞


≤1

0(q)

t

0(t − s)q−1µ(s)ds∥x1 − x2∥∞ + kλ∗

∥x1 − x2∥∞

≤1

0(q)

t

0((t − s)q−1)

11−γ ds

1−γ t

0(µ(s))

1γ ds

γ∥x1 − x2∥∞ + kλ∗

∥x1 − x2∥∞

≤1

0(q)

1 − γ

q − γ

1−γ

tq−γ T

0(µ(s))

1γ ds

γ∥x1 − x2∥∞ + mλ∗

∥x1 − x2∥∞

≤

1

0(q)

1 − γ

q − γ

1−γ

T q−γµ∗+ mλ∗

∥x1 − x2∥∞

= c∥x1 − x2∥∞. (19)

Combining (17), (18) and (19), we obtain

∥Ax1 − Ax2∥∞ ≤ c∥x1 − x2∥∞.

Since c < 1, it follows that A is a strict contraction. As a consequence of the Banach fixed point theorem, we deduce thatthere exists a unique fixed point which is a unique solution of the IVP (1) on J . The proof is complete. �

[H2′]: There exists a positive constant µ such that ∥f (t, φ) − f (t, ψ)∥ ≤ µ∥φ − ψ∥1, for almost every t ∈ J and allφ,ψ ∈ PC1.

Corollary 3.4. Assume [H1], [H2′] and [H3] hold, then the IVP (1) has a unique solution on J, provided that the inequality (8) and

the following inequality

c ′:=

10(q + 1)

T qµ+ mλ∗ < 1 (20)

hold.

Proof. According to condition [H2′] and inequality (20), for arbitrary x1, x2 ∈ PC1([−τ , T ])(ρ), for t ∈ [−τ , 0],

∥(Ax1)(t)− (Ax2)(t)∥ = ∥ϕ(t)− ϕ(t)∥ ≡ 0.

For t ∈ [0, t1],

∥(Ax1)(t)− (Ax2)(t)∥ ≤1

0(q)

t

0(t − s)q−1

[f (s, x1s )− f (s, x2s )]ds

≤1

0(q)µ

t

0(t − s)q−1ds sup

s∈[0,t]∥x1s − x2s ∥1

≤1

0(q + 1)tq1µ∥x1 − x2∥∞

≤ c ′∥x1 − x2∥∞.


∥(Ax1)(t)− (Ax2)(t)∥ ≤1

0(q)

t

0(t − s)q−1

[f (s, x1s )− f (s, x2s )]ds+

ki=1

∥Ii(x1(t−i ))− Ii(x2(t−i ))∥

≤1

0(q + 1)T qµ∥x1 − x2∥∞ + mλ∗

∥x1 − x2∥∞

≤

1

0(q + 1)T qµ+ mλ∗

∥x1 − x2∥∞

= c ′∥x1 − x2∥∞.

Since c ′ < 1, it follows that A is a strict contraction. As a consequence of the Banach fixed point theorem, we deduce thatthere exists a unique fixed point which is a unique solution of the IVP (1) on J . The proof is complete. �

4. Data dependence analysis

In this section, we study the data dependence of the solution of the IVP (1).


Theorem 4.1. Assume the condition of Theorem 3.3 hold. Let x(t) be the unique solution of (5), and let x(t) be the unique solutionof (5) satisfying the initial value x|[−τ ,0] = ϕ, where ϕ ∈ PC1. If for arbitrary ε > 0, ∥ϕ − ϕ∥1 < δ = (1 − c)ε, then we have∥x − x∥∞ < ε.

Proof. By Lemma 2.7, one can obtain that

x(t)− x(t) =

(ϕ(0)− ϕ(0))+1

0(q)

t

0(t − s)q−1(f (s, xs)− f (s, xs))ds, for t ∈ [0, t1],

(ϕ(0)− ϕ(0))+1

0(q)

t

0(t − s)q−1(f (s, xs)− f (s, xs))ds for t ∈ (tk, tk+1],

+

ki=1

(Ii(x(t−i ))− Ii(x(t−i ))), k = 1, . . . ,m.

(21)

For t ∈ [0, t1],

∥x(t)− x(t)∥ ≤ ∥ϕ(0)− ϕ(0)∥ +

t

0(t − s)q−1

∥f (s, xs)− f (s, xs)∥ds

≤ ∥ϕ − ϕ∥1 +

t


s∈[0,t]∥xs − xs∥1

≤ ∥ϕ − ϕ∥1 +1

0(q)

t

0((t − s)q−1)

11−γ ds

1−γ t

0(µ(s))

1γ ds

γ∥x1 − x2∥∞

≤ ∥ϕ − ϕ∥1 +1

0(q)

1 − γ

q − γ

1−γ

tq−γ1 µ∗∥x1 − x2∥∞

≤ ∥ϕ − ϕ∥1 + c∥x1 − x2∥∞. (22)

For t ∈ (tk, tk+1], k = 1, . . . ,m,

∥x(t)− x(t)∥ ≤ ∥ϕ(0)− ϕ(0)∥ +

ki=1

∥Ii(x(t−i ))− Ii(x(t−i ))∥ +

t

0(t − s)q−1

∥f (s, xs)− f (s, xs)∥ds

≤ ∥ϕ − ϕ∥1 + kλ∗∥x − x∥∞ +

t


s∈[0,t]∥xs − xs∥1

≤ ∥ϕ − ϕ∥1 + kλ∗∥x − x∥∞ +

10(q)

t

0((t − s)q−1)

11−γ ds

1−γ t

0(µ(s))

1γ ds

γ∥x − x∥∞

≤ ∥ϕ − ϕ∥1 + mλ∗∥x − x∥∞ +

10(q)

1 − γ

q − γ

1−γ

T q−γµ∗∥x − x∥∞

≤ ∥ϕ − ϕ∥1 + c∥x − x∥∞. (23)

Combining (22), (23) and note that ∥x(t)− x(t)∥ = ∥ϕ(t)− ϕ(t)∥ ≤ ∥ϕ − ϕ∥1 for t ∈ [−τ , 0], it implies

∥x − x∥∞ ≤ ∥ϕ − ϕ∥1 + c∥x − x∥∞,

c < 1 yields

∥x − x∥∞ ≤1

1 − c∥ϕ − ϕ∥1.

Therefore, if ∥ϕ − ϕ∥1 < δ, then ∥x − x∥∞ < ε. �

Corollary 4.2. Assume that all the conditions of Corollary 3.4 hold. Let x(t) be the unique solution of (5), and let x(t) be theunique solution of (5) satisfying the initial value x|[−τ ,0] = ϕ, where ϕ ∈ PC1. If for arbitrary ε > 0, ∥ϕ− ϕ∥1 < δ = (1− c ′)ε,then we have ∥x − x∥∞ < ε.

5. Example

Finally, we give some examples to illustrate the existence, uniqueness and data dependence results obtained in thispaper.


Example 1. Consider the following equation:

cD12 x(t) =

15sinxt −

π

2

, t ∈

0,

52π

, t =

kπ2,

1xkπ2

=

15, k = 1, 2, 3, 4,

x(t) = sin(t), t ∈

−π

2, 0.

(24)

Because of ∥15 sin(x(t −

π2 ))∥ ≤

15 , let m(t) ≡

15 , then m(t) ∈ L

14 ([0, 5

2π ]), we have β =14 and M = (

T0 (m(s))

1β ds)β =

( 5π

20

15

4ds)

14 =

15

5π2

14 . On the other hand, since ∥ϕ(0)∥ = 0,m = 4 andM∗

=15 , it is easy to know that

∥ϕ(0)∥ +1

0(q)

1−βq−β

1−βT q−βM + mM∗

ρ=

1√π

× 334 ×

15 ×

5π2

12 + 4 ×

15

2≈ 0.76 < 1

holds for ρ = 2. Then the IVP (24) at least has a solution on [0, 52π ] according to Theorem 3.1.

Example 2. We still consider the IVP (24).

cD12 x(t) =

15sinxt −

π

2

, t ∈

0,

52π

, t =

kπ2,

1xkπ2

=

15, k = 1, 2, 3, 4,

x(t) = sin(t), t ∈

−π

2, 0.

Since ∥15 sin(x(t− π

2 ))−15 sin(y(t− π

2 ))∥ ≤15∥x(t−

π2 )−y(t− π

2 )∥1,1x kπ

2

=

15 , k = 1, 2, 3, 4, we can let thatµ(t) ≡

15

and λ∗= 0, then µ(t) ∈ L

14 ([0, 5

2π ]), γ =14 and µ∗

=15

5π2

14 . Therefore

10(q)

1 − γ

q − γ

1−γ

T q−γµ∗+ mλ∗

=1

√π

× 334 ×

15

×

5π2

12

≈ 0.72 < 1

holds. The IVP (24) has a unique solution on [0, 52π ] according to Theorem 3.3.

On the other hand, the IVP (24) is also satisfy all the conditions of Theorem 4.1. Let x(t) be the unique solution of(24), and let x(t) be the unique solution of (24) satisfying the initial value x|[− π

2 ,0]= ϕ, where ϕ ∈ PC1. If for arbitrary

ε > 0, ∥ϕ − ϕ∥1 < δ = (1 − c)ε, then we have ∥x − x∥∞ < ε.

References

[1] D.D. Bainov, P.S. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, Longman Scientific and Technical Group. Limited,New York, 1993.

[2] V. Lakshmikantham, D.D. Bainov, P.S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, London, 1989.[3] M. Benchohra, J. Henderson, S.K. Ntouyas, Impulsive Differential Equations and Inclusions, vol. 2, Hindawi Publishing Corporation, New York, 2006.[4] K. Diethelm, The Analysis of Fractional Differential Equations, in: Lecture Notes in Mathematics, 2010.[5] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, in: North-HollandMathematics Studies, vol. 204,

Elsevier Science B.V., Amsterdam, 2006.[6] V. Lakshmikantham, S. Leela, J. Vasundhara Devi, Theory of Fractional Dynamic Systems, Cambridge Scientific Publishers, 2009.[7] K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York, 1993.[8] M.W.Michalski, Derivatives of noninteger order and their applications, DissertationesMathematicae, Polska AkademiaNauk., InstytutMatematyczny,

Warszawa, 1993.[9] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.

[10] V.E. Tarasov, Fractional Dynamics: Application of Fractional Ccalculus to Dynamics of Particles, Fields and Media, Springer, HEP, 2010.[11] R.P. Agarwal, M. Benchohra, S. Hamani, A survey on existence results for boundary value problems of nonlinear fractional differential equations and

inclusions, Acta Appl. Math. 109 (2010) 973–1033.[12] B. Ahmad, J.J. Nieto, Existence of solutions for anti-periodic boundary value problems involving fractional differential equations via Leray–Schauder

degree theory, Topol. Methods Nonlinear Anal. 35 (2010) 295–304.[13] Z.B. Bai, On positive solutions of a nonlocal fractional boundary value problem, Nonlinear Anal. TMA 72 (2010) 916–924.[14] M. Benchohra, J. Henderson, S.K. Ntouyas, A. Ouahab, Existence results for fractional order functional differential equationswith infinite delay, J. Math.

Anal. Appl. 338 (2008) 1340–1350.[15] G.M. Mophou, G.M. N’Guérékata, Existence of mild solutions of some semilinear neutral fractional functional evolution equations with infinite delay,

Appl. Math. Comput. 216 (2010) 61–69.


[16] J. Wang, Y. Zhou, Existence and controllability results for fractional semilinear differential inclusions, Nonlinear Anal. RWA 12 (2011) 3642–3653.[17] J.Wang, Y. Zhou,W.Wei, A class of fractional delay nonlinear integrodifferential controlled systems in Banach spaces, Commun. Nonlinear Sci. Numer.

Simul. 16 (2011) 4049–4059.[18] J. Wang, Y. Zhou, Analysis of nonlinear fractional control systems in Banach spaces, Nonlinear Anal. TMA 74 (2011) 5929–5942.[19] S. Zhang, Existence of positive solution for some class of nonlinear fractional differential equations, J. Math. Anal. Appl. 278 (2003) 136–148.[20] Y. Zhou, F. Jiao, Nonlocal Cauchy problem for fractional evolution equations, Nonlinear Anal. RWA 11 (2010) 4465–4475.[21] Y. Zhou, F. Jiao, J. Li, Existence and uniqueness for p-type fractional neutral differential equations, Nonlinear Anal. TMA 71 (2009) 2724–2733.[22] Y. Zhou, F. Jiao, J. Li, Existence and uniqueness for fractional neutral differential equations with infinite delay, Nonlinear Anal. TMA 71 (2009)

3249–3256.[23] A.M.A. El-Sayed, Nonlinear functional differential equations of arbitrary orders, Nonlinear Anal. TMA 33 (1998) 181–186.[24] V. Lakshmikantham, Theory of fractional functional differential equations, Nonlinear Anal. TMA 69 (2008) 3337–3343.[25] R.P. Agarwal, M. Benchohra, S. Hamani, A survey on existence results for boundary value problems of nonlinear fractional differential equations and

inclusions, Acta Appl. Math. 109 (2010) 973–1033.[26] M. Benchohra, S. Hamani, The method of upper and lower solutions and impulsive fractional differential inclusions, Nonlinear Anal. Hybrid Syst. 3

(2009) 433–440.[27] J. Wang, M. Feckan, Y. Zhou, On the new concept of solutions and existence results for impulsive fractional evolution equations, Dynam. Part. Differ.

Eq. 8 (2011) 345–361.[28] J. Wang, L. Lv, Y. Zhou, Ulam stability and data dependence for fractional differential equations with Caputo derivative, Electron. J. Qual. Theory Differ.

Equ. (63) (2011) e1–e10.[29] J. Wang, L. Lv, Y. Zhou, New concepts and results in stability of fractional differential equations, Commun. Nonlinear Sci. Numer. Simul. (2011),

doi:10.1016/j.cnsns.2011.09.030.[30] J. Wang, Y. Zhou, Mittag-Leffler–Ulam stabilities of fractional evolution equations, Appl. Math. Lett. (2011), doi:10.1016/j.aml.2011.10.009.[31] M. Feckan, Y. Zhou, J. Wang, On the concept and existence of solution for impulsive fractional differential equations, Commun. Nonlinear Sci. Numer.

Simul. (2011) doi:10.1016/j.cnsns.2011.11.017.

http://dx.doi.org/doi:10.1016/j.cnsns.2011.09.030

http://dx.doi.org/doi:10.1016/j.aml.2011.10.009

http://dx.doi.org/doi:10.1016/j.cnsns.2011.11.017

Documents

Impulsive fractional functional differential equations